Theorem spw 2038

Description: Weak version of the specialization scheme sp 2177. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2177 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2177 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2132 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2177 are spfw 2037 (minimal distinct variable requirements), spnfw 1984 (when 𝑥 is not free in ¬ 𝜑), spvw 1985 (when 𝑥 does not appear in 𝜑), sptruw 1809 (when 𝜑 is true), spfalw 2002 (when 𝜑 is false), and spvv 2001 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1914	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1914	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1914	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2037	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783

This theorem is referenced by:  hba1w  2051  19.8aw  2054  exexw  2055  spaev  2056  ax12w  2130  nfcriOLD  2894  rspw  3234  reldisj  4452  ralidmw  4508  dtruALT2  5369  dtruOLD  5442  bj-ssblem1  35531  bj-ax12w  35554